On the coefficients of a typically-real function

The Critical Points of a Typically-Real Function

“…Let z2=x+iy, zx=-x+iy where x>0 andy>0. M. S. Robertson [7] proved that each function in TR has a Stieltjes integral representation <i8> ™-kÍ, /*"!+.-f*»- _ f*(Z| -zi)(l -zxz2) dp(B)…”

“…(1) /(*) = z + 2 a"*"> regular in F:|z|<l, is said to be typically-real if it satisfies the condition (2) (3/(z))(3z) > 0 for all nonreal z in E. This class of functions (which we denote by TR) was introduced by Rogosinski [9] in 1932 and has been the object of many investigations ( [2], [3], [7]). The condition (2) implies that f(z) is real in E, if and only if z is real.…”

“…The work closes with a theorem on the domain of univalence of the class TR and a conjecture on the domain of fc-valence of the class TR. Our main tool is the theory of subordination, but we also use a Stieltjes integral representation due to M. S. Robertson [7].…”

Families of real and symmetric analytic functions

“…The class Q was introduced by M. S. Robertson [7]. The condition that / "preserves quadrants" implies that /(z) is real when z is real and / is odd (as we verify later).…”

“…For the class Q2, Robertson [7] proved that \a3\ < 1 and, in general, |a2n_,| + \a2n+ il ** 2. For the family of odd, typically real functions the coefficents are much less restrictive than for Q2 and the bound |fl2n+,| < 2n + 1 [8] is the sharp result.…”